Quoting Due w/ a little boy!:" Yes those were the ones."
Breaking it down.
Once the excitement of the new semester
Once: Complement establishing time.
"The excitement": Subject of verb
"of the new semester": Adjunct, (complement specifying excitement)
"Had subsided": Verb
[the students began studying the material with less and less frequency
"The students" : Subject of verb
Began studying: Verb
The material: Direct object complement of verb
With less and less frequency: Adjunct, also establishes time (kinda, I can't come up with a better word than time)
So the highlighted things are both adjuncts basically, and an adjunct is something that is not necessary for the clause to function, they're just like extra info.